Kerodon

Remark 11.10.1.14 (Pullback Stability). Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cocartesian fibrations of simplicial sets, and let $f: X \rightarrow X'$ be an equivalence of cocartesian fibrations over $S$. For any morphism of simplicial sets $T \rightarrow S$, the induced map $f_{T}: T \times _{S} X \rightarrow T \times _{S} X'$ is an equivalence of cocartesian fibrations over $T$ (this follows immediately from Remark 11.10.1.9). In particular, for every vertex $s \in S$, the induced map of fibers $f_{s}: X_{s} \rightarrow X'_{s}$ is an equivalence of $\infty $-categories.